Courtyard sky view and diffuse light

Interactive model

See how the sky reaches the floor.

Adjust the courtyard proportions and wall height, then move the selected point across the floor. Compare its local sky view with the entire-floor average.

Area-averaged sky view factor decreases with wall height; the square is above the four-to-one and eight-to-one curves. — Isotropic diffuse transmission through the full opening to the entire floor. Download figure (SVG)

Enable JavaScript to explore the model. The published comparison above and results below remain available.

Colour shows calculated local sky view under an isotropic sky. The marker identifies the selected floor point. The area average comes from the published dataset; it is not the centre-point value. Fixed 100 m² floor, with no reflections.

Results

What the analysis shows

At a wall height of 5 m, the square courtyard has a sky view factor of 41.5%, compared with 34.6% for the 4:1 plan. The reduction is 6.9 percentage points. Orientation has no effect under an isotropic sky.

Wall height	Square	4:1 plan	8:1 plan
1 m	82.7%	78.7%	73.6%
2.5 m	63.2%	56.2%	48.6%
5 m	41.5%	34.6%	28.1%
7.5 m	28.3%	23.2%	18.6%
10 m	20.0%	16.5%	13.4%

Within the sampled family, increasing depth and elongation reduce diffuse access. The quantity represents the entire floor, not a view measured at one point.

Methods and study design

The horizontal floor and its equally sized, parallel top opening define a radiation view-factor problem. For an isotropic sky without reflections, the area-averaged irradiance transmission equals the floor-to-opening view factor. We evaluate 35 configurations: seven aspect ratios (1, 1.5, 2, 3, 4, 6, 8) and five wall heights (1, 2.5, 5, 7.5, 10 m), always at 100 m² area.

The primary calculation uses the closed-form equal-rectangle relationship catalogued by Howell (C-11). Two numerical formulations test it without reusing that algebraic expression:

Integrate the direct sunlit-area fraction across the sky hemisphere, weighted by the horizontal cosine and solid angle.
Integrate the floor-to-opening radiative kernel over relative horizontal displacements.

Fsky = (1 / π) ∫∫ f(α, a) sin(α) cos(α) dα da Fsky = [4 / (π W L)] ∫₀ᵂ ∫₀ᴸ (W − x)(L − y) H² / (H² + x² + y²)² dy dx

Local sky-view display

The interactive floor map integrates the same floor-to-opening kernel at each selected position. For a point (x, y), the integrand over the opening is H² / [π(H² + Δx² + Δy²)²]. Its rectangular antiderivative is evaluated at the four opening corners. The heatmap samples 28 × 28 floor positions; its colour field is a spatial display, while the reported area average uses the published closed-form dataset.

The selected point can differ substantially from the floor average. The browser implementation is provided with the research materials for inspection.

Both midpoint quadratures are repeated at 60, 120 and 240 subdivisions per axis for five configurations spanning shallow, deep, square and elongated geometries. A substitution u = sin²(α) makes the hemispherical cosine weighting explicit. Symmetry permits a single azimuth quadrant.

At the finest resolution, maximum absolute differences from the closed form are 0.0000344 for the hemispherical integral and 0.0000135 for the area kernel. These are internal numerical comparisons across different formulations.

Interpretation and scope

The opening’s area is fixed, but its distribution relative to each point on the floor changes. Elongation brings more of the floor close to a long wall; height increases the obstruction angle. The radiative integral captures both effects.

The hemispheric and area-kernel approaches provide a reusable cross-check for this class of models. The view-factor relationship itself is established, not an ABLAQ invention. An overcast or clear real sky can be anisotropic; reflected light from walls and multiple reflections are also excluded. A centre-point sky-view estimate must not be substituted for this area-averaged result.

Data and reproducibility

The interactive model uses the published data and open browser equations (JavaScript). The executed interactive-model verification records numerical agreement and integration checks. Its interface source is also available. The controls run locally; shared links encode only the chosen model inputs.

All numerical outputs are calculated, not observational measurements. The release contains the complete parameter tables, source and numerical comparison record.

Code, numerical data and generated research figures are released under CC0 1.0. The report is institutionally published; numerical verification does not constitute external peer review.

Methodological references

NOAA Global Monitoring Division, General Solar Position Calculations. Fractional-year declination approximation and solar hour-angle relationships.
John R. Howell, A Catalog of Radiation Heat Transfer Configuration Factors, C-11. Equal, parallel, directly opposed rectangular surfaces.
EnergyPlus Engineering Reference, Shading Module. Geometric treatment of solar shading.
Sandia National Laboratories, PV Performance Modeling Collaborative: Plane of Array Irradiance. Distinguishing beam, diffuse-sky and ground-reflected components.

Institutional authorship

Author and publisher: ABLAQ Studios. Osama Khaled Ahmed Abdel-Qader, Scientific Research Lead, directs the scientific research and its decisions and holds publication accountability. Current team responsibilities are described in the research directory; programme roles are not a substitute for study-specific contribution records.

Computational implementation and execution use the tools documented in Methods. Scientific correspondence: osama.khaled@ablaqstudios.com.

ABLAQ Studios (2026). Courtyard sky view and diffuse light. ABLAQ-R03, version 1.0.0. https://ablaqstudios.com/sky-view-and-diffuse-light. Download citation (BibTeX)